EQUATION OF ENERGY TRANSPORT IN
ONE-DIMENSIONAL SEMI-INFINITE DOMAINS
VLADIMIR V. KULISH
Received 23 March 2004 and in revised form 28 April 2004
This paper presents an integral solution of the generalized one-dimensional equation of energy transport with the convective term. The solution of the problem has been achieved by the use of a novel technique that involves generalized derivatives (in particular, deriva- tives of noninteger orders). Confluent hypergeometric functions, known as Whittaker’s functions, appear in the course of the solution procedure upon applying the Laplace transform to the original transport equation. The analytical solution of the problem is written in the integral form and provides a relationship between thelocalvalues of the transported property (e.g., temperature, mass, momentum, etc.) and its flux. The solu- tion is valid everywhere within the domain, including the domain boundary.
1. Introduction
This paper presents a further development of the method that was first discussed by Old- ham and Spanier [12]. The same method was successfully used in numerous applications [4,5,6]. In recent works, the method was extended and applied to problems involving combustion [7], hyperbolic heat transfer [10,11], turbulent flows [2], and such problems in biomedical engineering as modeling of the neural response to an external stimulus [9]
and the alveolar gas exchange [3].
This work is an attempt to generalize the above-mentioned method.
2. Generalized equation of energy transport
A generalized energy transport equation with the convective term can be obtained from the conservation equation written in the differential form
∂F(r,t)
∂t +∇ ·ϕ(r,t)=S(r,t), (2.1) whereFrepresents the transported property (such as, temperature, mass concentration, momentum, etc.),ϕdenotes the flux ofF,S(r,t) is the source function, whereasrandt are the spatial and time-independent variables, respectively.
Copyright©2004 Hindawi Publishing Corporation Mathematical Problems in Engineering 2004:3 (2004) 185–195 2000 Mathematics Subject Classification: 35L05, 44A10, 80A20, 26A33 URL:http://dx.doi.org/10.1155/S1024123X0440307X
The conservation equation involves two unknown variables,Fandϕ, and hence must be coupled with a constitutive equation that would relate these unknown quantities.
In general, constitutive equations are but assumptions and, unlike the conservation equation, cannot be derived from fundamental principles. Therefore, the final form of the equation of energy transport depends on the form of the constitutive equation used.
In the present model, the following constitutive equation is used:
ϕ(r,t+τ)= −D(r,t)∇F(r,t) +u(r,t)F(r,t), (2.2) whereDis the diffusion coefficient always measured in m2/s (it can be, e.g., mass diffu- sivity, kinematic viscosity, or the coefficienth/mthat appears in the Schr¨odinger equa- tion, wherehis Planck’s constant andmis the particle’s mass) andu(r,t) denotes the velocity vector—therefore, quantityFis transported by both diffusion (the first term in the right-hand side of the constitutive equation) and convection (the second term in the right-hand side). The parameterτrepresents the time lag between the onset of the gra- dient of the transported quantity and the occurrence of the flux of that quantity. Hence, unlike Fick’s or Fourier’s constitutive equations, (2.2) accounts for afinite speedof the transport process and is more general than those equations.
The left- and right-hand sides of the constitutive relation are written for two different time moments. In order to overcome this difficulty, the left-hand side of (2.2) is expanded into the Taylor series. The constitutive equation becomes
∞ n=0
τn n!
∂nϕ(r,t)
∂tn = −D(r,t)∇F(r,t) +u(r,t)F(r,t), (2.3) where∂0ϕ(r,t)/∂t0=ϕ(r,t).
Upon applying the divergence operator to both parts of (2.3), the latter becomes ∞
n=0
τn n!
∂n∇ ·ϕ(r,t)
∂tn = −∇ ·
D(r,t)∇F(r,t)+∇ ·
u(r,t)F(r,t). (2.4) Now it follows from (2.1) that∇ ·ϕ(r,t)= −∂F(r,t)/∂t+S(r,t). Upon substituting this into (2.4) and rearranging the terms, the equation becomes
∞ n=0
τn n!
∂n+1F(r,t)
∂tn+1 +∇ ·
u(r,t)F(r,t)
= ∇ ·
D(r,t)∇F(r,t)+S(r,t) + ∞ n=1
τn n!
∂nS(r,t)
∂tn .
(2.5)
Equation (2.5) is the generalized equation of energy transport. It reduces to the classical diffusion (heat) equation ifτ=0 andu(r,t)=0. Ifu(r,t)=0 andτ1 (all the terms whose order is larger than one can be neglected in the series), the generalized equation of energy transport reduces to the classical wave equation, whereasc=√
D/τ is the propa- gation speed of waves (e.g., speed of light or sound). Note the presence of the apparent energy source∞n=1(τn/n!)(∂nS(r,t)/∂tn) in (2.5); it appears due to the finite time lag be- tween the excitation and the response to it.
If the diffusion coefficient and velocity are both constant, that is, D(r,t)=D and u(r,t)=u, (2.5) becomes
∞ n=0
τn n!
∂n+1F(r,t)
∂tn+1 +u∇F(r,t)=D∇2F(r,t) +S(r,t) + ∞ n=1
τn n!
∂nS(r,t)
∂tn . (2.6) 3. Problem formulation
Consider an energy transport process that occurs in a homogeneous (no preferred direc- tion), semi-infinite domain whose boundary moves with a constant speeduand whose diffusion coefficient depends neither on spatial variable nor on time. In this case, the generalized equation of energy transport becomes
∞ n=0
τn n!
∂n+1F(r,t)
∂tn+1 +u∂F(r,t)
∂r
=D
∂2F(r,t)
∂r2 +2γ r
∂F(r,t)
∂r
+S(r,t) + ∞ n=1
τn n!
∂nS(r,t)
∂tn .
(3.1)
The parameterγcharacterizes the domain geometry. Thus,γ=0 corresponds to the do- main with the flat boundary (no curvature);γ= ±1 represents the spherical case with the convex and the concave boundary, respectively; whereasγ= ±1/2 describes the cylinder whose boundary is either convex (γ=1/2) or concave (γ= −1/2).
The spatial variabler=x±R, wherexis the actual distance from the origin andRis the initial radius of curvature. Note that the sign ofRmust be the same as the sign ofγ.
Initially, att=0, the domain is in equilibrium with a constant value of the transported quantity,F0, throughout the domain, 0≤r <+∞.
As the energy transport process goes on, the condition limr→∞F(r,t)=F0 must be imposed in order to comply with the principle of energy conservation.
At this point, the second boundary condition is deliberately not imposed. This issue will be clarified in the following section.
4. Solution procedure
Upon introducing the new variableρ=r/√Dand the excess of the transported quantity Fˆ=F−F0, the transport equation becomes
∞ n=0
τn n!
∂n+1F(ρ,t)ˆ
∂tn+1 =
∂2F(ρ,t)ˆ
∂ρ2 + 2 γ
ρ−ω
∂F(ρ,t)ˆ
∂ρ + ∞ n=0
τn n!
∂nS(ρ,t)
∂tn , (4.1) whereω=u/2√Dand∂0S(ρ,t)/∂t0=S(ρ,t).
The initial condition becomes ˆF(ρ, 0)=0 and the boundary condition is now limρ→∞F(ρ,t)ˆ =0.
Upon taking the Laplace transform of (4.1), the latter becomes d2Φ(ρ;s)
dρ2 + 2 γ
ρ−ω
dΦ(ρ;s)
dρ −sesτΦ(ρ;s)= −esτQ(ρ,s), (4.2)
whereΦis the Laplace transform of the excess of the transported quantity ˆF,sis the Laplace transform variable, andQ(ρ,s) represents the Laplace transform of the source functionS, provided that this Laplace transform exists.
The general solution of (4.2) is
Φ(ρ;s)= C1(s)Mκ,µ2ρ f(s)+C2(s)Wκ,µ2ρ f(s)eωρ
ργ +P(ρ,s), (4.3) where f(s)=√
sesτ+ω2,P(ρ,s) is a particular solution of (4.2),C1(s) andC2(s) are two arbitrary functions of the Laplace transform variables,Mκ,µ(z) andWκ,µ(z) are Whit- taker’s functions defined as
Mκ,µ(z)=e−z/2zµ+1/2M 1
2+µ−κ, 1 + 2µ,z
, Wκ,µ(z)=e−z/2zµ+1/2U
1
2+µ−κ, 1 + 2µ,z
,
(4.4)
wherez=2ρ f(s),κ=γω/ f(s), andµ=γ−1/2. Note that the value ofzincreases asρ increases. FunctionsMandUin (4.4) are Kummer’s confluent hypergeometric functions defined as
U(a,b,z)= π sin(πb)
M(a,b,z)
Γ(1 +a−b)Γ(b)−z1−bM(1 +a−b, 2−b,z) Γ(a)Γ(2−b)
M(a,b,z)=1 +az
b +(a)2z2
(b)22!+···+(a)nzn (b)nn!+···,
(4.5)
where (a)n=a(a+ 1)(a+ 2)···(a+n−1), (b)n=b(b+ 1)(b+ 2)···(b+n−1), and (a)0=(b)0=1 (see [1, page 504]).
Written in terms of Kummer’s functions, (4.3) becomes Φ(z;s)=
C1(s)M(a,b,z) +C2(s)U(a,b,z)2f(s)γe−z(1+ω/ f(s))/2+P(z,s), (4.6) wherea=γ(1−ω/ f(s)) andb=2γ.
Now, since lim|z|→∞M(a,b,z)=(Γ(b)/Γ(a))ezza−b(see [1, page 504]), the first term in (4.6) becomes unbounded for large values ofz. This, however, contradicts the bound- ary condition limz→∞Φ(z;s)=0. Hence, for the solution to be bounded, the arbitrary functionC1(s) must be identically zero. On the other hand, lim|z|→∞U(a,b,z)=z−a(see [1, page 504]). Therefore, provided thats >0 (this is really the case because the Laplace transform variable corresponds to time and is always positive), the second term in (4.6) decreases aszincreases and vanishes aszbecomes infinitely large. Consequently, the so- lution becomes
Φ(ρ;s)=C(s)Ua,b, 2ρ f(s)2f(s)γeρ[ω−f(s)]+P(ρ,s). (4.7)
Upon differentiating (4.7) with respect toρ, the equation becomes dΞ
dρ =C(s)2f(s)γ −2a f(s)Ua+ 1,b+ 1, 2ρ f(s) +Ua,b, 2ρ f(s)ω−f(s)eρ[ω−f(s)],
(4.8)
whereΞ(ρ;s)=Φ(ρ;s)−P(ρ;s). Note that dUa,b, 2ρ f(s)
dρ = −2a f(s)Ua+ 1,b+ 1, 2ρ f(s). (4.9) Furthermore, it follows from (4.7) that
C(s)= 1
Ua,b, 2ρ f(s)Ξ(ρ;s)2f(s)−γe−ρ[ω−f(s)]. (4.10) Upon substituting this into (4.8), the latter reduces to
dΞ dρ=
ω−f(s)+ 2γω−f(s)Ua+ 1,b+ 1, 2ρ f(s) Ua,b, 2ρ f(s)
Ξ(ρ;s). (4.11)
Dividing (4.11) by f(s),
− 1 f(s)
dΞ dρ
=
1− ω f(s)+ 2γ
1− ω
f(s)
Uγ1−ω/ f(s)+ 1, 2γ+ 1, 2ρ f(s) Uγ1−ω/ f(s), 2γ, 2ρ f(s)
Ξ(ρ;s).
(4.12)
Note the minus sign in the left-hand side.
Now, upon noticing thatU(a+ 1,b+ 1,z)/U(a,b,z)=1/z=1/[2ρ f(s)], (4.12) simpli- fies to
− 1 f(s)
dΞ dρ =
1 +
γ ρ−ω
1 f(s)−
γω ρ
1 f2(s)
Ξ(ρ;s). (4.13)
In most practical applications, the values of the time lagτare very small. In fact,τ= D/c2, wherecis the speed of the energy-transporting waves (it is usually very large, while the diffusion coefficientDis small). Hence, in this case, f(s)=√
sesτ+ω2can be written as f(s)=
s(sτ+ 1) +ω2. The inverse Laplace transform of 1/ f(s) is [1, page 1025]
L−1
1 s(sτ+ 1) +ω2
= 1
√τL−1
1 (s+ 1/2τ)2−
1/4τ2−ω2/τ
=e−t/(2τ)
√τ I0
t 2τ
1−4τω2
,
(4.14)
whereI0is the modified Bessel function. The inverse Laplace transform of 1/ f2(s) is [1, page 1022]
L−1
1 s(sτ+ 1) +ω2
=1 τL−1
1 (s+ 1/2τ)2−
1/4τ2−ω2/τ
= 2e−t/(2τ)
√1−4τω2sinh t
2τ
1−4τω2
.
(4.15)
Upon taking the inverse Laplace transform of (4.13) and restoring the original vari- ables, the solution becomes
F(r,t)=F0−c t
0e−(t−ζ)/(2τ)I0
t−ζ 2τ
1−u2
c2
∂F(r,ζ)−p(r,ζ)
∂r dζ
+ u
2τc− γc
r t
0e−(t−ζ)/(2τ)I0
t−ζ 2τ
1−u2
c2
F(r,ζ)−F0−p(r,ζ)dζ
+ γu
r√1−u2/c2 t
0e−(t−ζ)/(2τ)sinh t−ζ
2τ
1−u2 c2
F(r,ζ)−F0−p(r,ζ)dζ +p(r,t),
(4.16) wherep(r,t) is the inverse Laplace transform of the particular solutionP(r;s).
Finally, upon substituting the constitutive equation relating the transported quantity Fwith its fluxφ, namely,−∂F(r,t)/∂r=(1/D)[φ(r,t) +τ(∂φ(r,t)/∂t)−uF(r,t)], the so- lution becomes
F(x,t)=F0+1
t
0e−(t−ζ)/(2τ)I0
t−ζ 2τ
1−u2
c2
φ(x,t)+τ∂φ(x,ζ)
∂t −uF(x,ζ)+β(x,ζ)
dζ +
u 2−
γc x±R
t
0e−(t−ζ)/(2τ)I0
t−ζ 2τ
1−u2
c2
F(x,ζ)−F0−p(x,ζ)dζ
+ γu
(x±R)√1−u2/c2 t
0e−(t−ζ)/(2τ)sinh t−ζ
2τ
1−u2 c2
F(x,ζ)−F0−p(x,ζ)dζ +p(x,t),
(4.17) whereβ(x,t)= −D∂p(x,t)/∂xdenotes the effective flux due to the presence of the source function in the original equation, andRrepresents the radius of curvature of the surface.
The sign ofRmust be the same as the sign of the parameterγ. The parameter=cτ is the “space quantum.”
Equation (4.17) is an integral solution of the generalized problem of energy transport modeled by (3.1) in the case of small values of the phase lag τ. It provides a relation- ship between thelocal values of the transported quantityF and its fluxφand is valid everywhere within the domain, including the domain boundary.
5. Some special cases of the solution
In some special cases, the solution given by (4.17) must reduce to some known solutions of the transport problem. In this section, it is shown that this is really the case.
Note that limz→∞I0(z)=ez/√2πz[1, page 377].
Hence, in the case ofτ=0 andu=0, (4.17) reduces to F(x,t)=F0+∂−1/2φ(x,t)−β(x,t)
∂t−1/2 +p(x,t) + γ
x±R
√D∂−1/2p(x,t)−F(x,t) +F0
∂t−1/2 ,
(5.1)
where the derivatives of a negative fractional orderαare defined as dαf(t)
dtα = 1 Γ(−α)
t
0
f(τ)dτ
√t−τ (5.2)
for any f(t) for which the above integral exists.
The solution given by (5.1) has been reported in the case of no source function in [12].
Furthermore, ifγ=0 in (5.1)—which corresponds to the planar geometry—the solu- tion becomes identical to the solution reported in [6]. If the source function is absent in the original equation, then (5.1) withγ=0 becomes the case reported in [4,5].
On the other hand, ifγ=0,u=0, and no source function is present in (4.17), the solution coincides with the integral equation reported in [7].
Finally, in the case ofu=0, the solution given by (4.17) becomes F(x,t)=F0+1
t
0e−(t−ζ)/(2τ)I0
t−ζ 2τ
φ(x,t) +τ∂φ(x,ζ)
∂t +β(x,ζ)
dζ
− γc x±R
t
0e−(t−ζ)/(2τ)I0
t−ζ 2τ
F(x,ζ)−F0−p(x,ζ)dζ+p(x,t).
(5.3)
Note that, in the case ofγ=0 and no source function, the solution is identical to the integral equation reported in [8].
6. Model validation
To validate the model, (4.17) has been numerically solved for various sets of parameters.
The physical properties of the domain were set as follows:c=103m/s andD=10−5m2/s.
Such a choice was made in order to be consistent with the results obtained in [8].
Figure 6.1 shows the normalized value of the transported quantity, F/Fmax, on the moving planar boundary for different values of the boundary speed, when the flux is constant (φ=100 units/m2). The solution for the latter case is well known,F/Fmax∼√
t.
One can see fromFigure 6.1that the stronger the convective effect is the slower the solu- tion grows.
Figure 6.2presents the case of an expanding sphere (R=4×10−8m). The evolution of the normalized transported property is shown on the boundary of the sphere, provided
×10−11 6 5 4
3 2 1 0
t, s
−0.2 0 0.2 0.4 0.6 0.8 1
F/Fmax
u=0 u=0.1c u=0.5c u=0.999c
Figure 6.1. Normalized value of the transported quantity in the case of a semi-infinite domain with the moving boundary (constant flux).
×10−11 6 5 4 3 2 1 0
t, s 0
0.2 0.4 0.6 0.8 1
F/Fmax
u=0 u=0.1c u=0.5c u=0.999c
Figure 6.2. Normalized value of the transported quantity in the case of an expanding sphere (constant flux).
that the surface flux is constant (φ=100 units/m2). The parameterscandDin the trans- port equation were set the same as in the previous case. The solution is further compared with the case of a stationary sphere (u=0).
Figure 6.3shows the evolution of the normalized transported quantity on the moving boundary (u=0.5c) in the case of a Gaussian flux. The flux was modeled as the surface fluxφsurf(t)=exp[−((t−b)/σ)2], withb=10 picoseconds andσ=5 picoseconds, that mimics an almost instantaneous source of energy. Such a choice was made in order to provide a comparison with the results obtained previously for the laser pulse heating [8].
×10−11 6 5 4 3 2 1 0
t, s 0
0.2 0.4 0.6 0.8 1
F/Fmax
g=0 (planar) g=0.5 (cylinder) g=1 (sphere)
Figure 6.3. Normalized value of the transported quantity in the case of a Gaussian flux.
×10−11 6 5 4 3 2 1 0
t, s 0
0.2 0.4 0.6 0.8 1
F/Fmax
u=0 u=0.1c u=0.5c u=0.999c
Figure 6.4. Normalized value of the transported quantity in the case of an expanding sphere (Gauss- ian flux).
It is evident from Figure 6.3that for the given set of conditions, the fastest energy transport takes place in the case of the spherical geometry. Yet all the solutions converge to each other as time increases. This is easily explained from the fact that the radius of the surface curvature increases as time goes on, limt→∞R= ∞, which corresponds to the planar geometry.
Finally,Figure 6.4shows the results obtained in the case of an expanding sphere (R= 4×10−8m) with the Gaussian surface flux (the same as in the preceding case). The effect of convection is clearly seen in the figure.
7. Conclusions
The paper presents a further development and generalization of the method that allows obtaining analytical solutions of various energy transport problems. The method is based on a technique that involves the use of generalized derivatives (sometimes reducible to derivatives of noninteger orders, known asfractional differ-integrals).
In this paper, the generalized equation of energy transport has been derived by cou- pling the conservation equation with the constitutive equation that represented the flux as the superposition of the classical Fick’s (Fourier’s) term and convective term. The con- stitutive equation also accounted for a possible finite time lag between the onset of the gradient of the transported quantity and its flux. The equation of energy transport thus obtained was then solved in its one-dimensional form within a semi-infinite domain whose geometry was governed by one of the parameters in the equation. The solution has been written in the form of convolution (memory) integrals that relate thelocalval- ues of the transported quantity and its flux. Such solutions are valid everywhere within the domain, including the domain boundary.
Furthermore, it has been shown that, in many special cases, the solution reduces to less general solutions that have been reported previously.
To validate the model, various solutions of the resulting integral equation have been numerically obtained. In the case of the planar geometry, all these solutions coincide with the solutions obtained in previous works. Furthermore, it has been shown that the solutions obtained in the case of the spherical geometry converge to the corresponding planar solutions as the radius of the surface curvature increases.
In addition, the solution given by (4.17) provides some important clues of how en- ergy transport processes take place in general. Thus, for instance, it follows that a certain maximal speed of energy transport should exist such that no process of energy transport may occur with a speed larger than that maximum speed (the parametercand the term
√1−u2/c2 in the solution). Furthermore, although allowed being very small, the time lagτin the solution is finite. This time lag may be viewed as the “time quantum” of the process in question. Moreover, the solution contains the “space quantum”defined as =cτ.
Curiously enough, in the case of a nonzero value of the velocityu, the solution given by (4.17) becomes a mapping of the formFn+1= (Fn), where denotes the integral operator in (4.17). Therefore, the solution of the generalized equation of energy transport allows chaotic or even biotic (self-organized) solutions [13]. This may become the topic of future studies.
References
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[2] V. V. Kulish,Fractional solutions for near-wall turbulence production, Fractional Differentiation and Its Applications, Bordeaux, 2004.
[3] ,Modeling of gas exchange in a single alveolus: mass transfer in an expanding sphere, 15th International Symposium on Transport Phenomena, Bangkok, 2004.
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[8] ,The relationship between the local temperature and the local heat flux within a one- dimensional semi-infinite domain of heat wave propagation, Math. Probl. Eng.2003(2003), no. 4, 173–179.
[9] , On the integral equation for the neural response to an external stimulus, 5th Inter- national Conference on Mathematical Problems in Engineering & Aerospace Sciences, Timisoara, 2004.
[10] V. V. Kulish and A. I. Sourin,Simulation and visualization of thermal wave propagation in pico- scales, 10th International Symposium on Flow Visualization, Kyoto, 2002.
[11] ,Simulation and visualization of thermal wave propagation in sub-nano-scales: ultra-fast laser heating of solid materials, 7th Asian Symposium on Visualization, Singapore, 2003.
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Vladimir V. Kulish: School of Mechanical and Production Engineering, Nanyang Technological University, Singapore 639798
E-mail address:[email protected]
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